Chemistry Reference
In-Depth Information
Fig. 2.6
For a process that
takes place on cooling from
T
0
to
T,
the flexible methods
estimate
E
ʱ
from the area
S
(
T
0
)−
S
(
T
) that corresponds
to the actually accomplished
extent of conversion. The
rigid methods estimate
E
ʱ
from
S
(
T
) that represents the
conversion, which is yet to be
accomplished
S(T
0
)-S(T)
S(T)
T T
0
integral method means that respective integration is performed assuming that the
process temperature is equal to the furnace temperature. This issue can be resolved
in a flexible integral method by replacing integration over temperature with inte-
gration over time. A specific method has been proposed by Vyazovkin [
35
]. In this
method, the aforementioned algorithmic solution (Eqs. 2.15, 2.16 and 2.17) is ad-
justed to integration over time. The
E
ʱ
value is then determined by minimizing the
following function [
35
]:
n
n
JE Tt
JE Tt
[, ( ]
[, ( ]
,
∑
1
∑
α
i
α
Φ()
E
=
(2.18)
α
i
=
ji
≠
α
j
α
where
J
[
E
ʱ
,
T
(
t
ʱ
)] is defined as:
t
a
E
RT t
−
()
=
∫
JE Tt
[,
] xp
α
d
t
(2.19)
α
α
()
0
and
T
i
(
t
) is a set of arbitrary temperature programs. The integral is solved numeri-
cally and minimization is repeated for each value of
ʱ
to obtain a dependence of
E
ʱ
on
ʱ
. Equation 2.19 affords substitution of the actual sample temperatures for
T
(
t
).
The use of the actual sample temperature in Eq. 2.19 makes the method applicable
to any preset program (i.e., isothermal as well as nonisothermal, be it linear or not)
even if it is distorted by the thermal effect of the process.
The fourth built-in assumption is that at any
ʱ,
the value of
E
ʱ
is estimated as
being constant in the whole range from 0 to
ʱ
. This again arises from the fact that
the traditional rigid integral methods estimate the temperature integral in the form
of Eq. 2.9, i.e., from 0 to
T
. As a result of such integration, the estimated
E
ʱ
values
are strictly accurate only when
E
ʱ
does not vary with
ʱ
. Otherwise, the
E
ʱ
estimates
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